We say that a Borel set \(P\)  is polar if there is a subharmonic function \(u \lnot \equiv -\infty \)  defined in a neighbourhood of \(P\)  so that \(P \subset \lbrace z: u(z) = -\infty \rbrace \) . For a domain \(\Omega \subset \mathbb {C}\) , a point \(\zeta _0 \in \partial \Omega \)  is said to be irregular if there is a \(\zeta \in \Omega \)  such that \(\lim _{z \rightarrow \zeta _0} G(z, \zeta ) \ne 0\)  and regular otherwise. By Kellogg's Theorem, cf. [1]}, the set of irregular boundary points of a domain is a polar set. The Green's function on a Riemann surface induces the logarithmic capacity, which is used prominently (see [1]} for its applications) in potential theory. Ahlfors introduced the analytic capacity for domains [3]}, [4]} in order to study Painlevé's question: which compact sets \(E\)  in the complex plane are removable for the bounded holomorphic functions?
